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This one is a bit different as the battery can be swapped out but the only way to charge these is to plug in the usb cable into the camera and into a usb port on your computer. The computer has to be running.and the camera has to be turned off. You will see a green light that shows it charging when the battery is charged the light will go out. Depending on how much power the port supplies it can take 3-8 hours. As long as the port is supply power it really doesn't matter what computer it is on as you are not transferring picstures so don't worry about plugging it into any thing that running to get a charge.

You've likely jammed the zoom motor in the lens. You're correct in thinking that the camera can't work without the lens out. With it being physical damage repairs wouldn't be covered under warranty so you either need to pay to have it repaired (if that's even possible, it varies camera to camera) or buy a new camera unfortunately.

The problem is that the lens has become stuck in the barrel. There are some DIY solutions you could try, but the
probability is that you will have to get it fixed by a professional.

Use
these at own risk as it may further damage the camera.

Firstly ,
try connecting your ac adapter or usb cable.

Try holding the
shutter button while switching on the camera.

Look at the lens ,
and if some of the lens 'circles' is misaligned or not concentric then
try wiggling it (while holding camera lens down).

Try gently
pushing or pulling the lens when it extends but this is risky as it may
cause the lens barrel to slip out of its guidance system.

Another
way to do this is to place the camera lens down on a hard surface and
then power it up. Be sure to use a soft cloth or something similar as to
not scratch your lens or casing. Let the lens push the camera up and
down a few times and sometimes the little resistance provided by the
camera is enough to get things going again.

Try hitting your
camera near the lens on the body with the soft tissue on the palm of
your hand.

Other than that , I would take the camera to a repair
center for a evaluation to see if it would cost more to repair than to
replace the camera.

If it is still under warranty I would suggest
you take it in before trying any of these steps and remove any
off-brand batteries or accessories as some stores are really fussy about
warranty repairs on camera's with non-brand accessories.

hopefully you haev installed the software first on yoy computer>
turn off the computer. connect the phone line one end to the ethernet port on your computer and the other end to one of the LAN port on the x6.
power up the (power cube shipped with the X6), the Pwr and the WLAN should come on and become steady, the LINK light should blink,
turn on you computer, connect a new phone line between ADSL port and the wall jack, the LINK light should become steady.